Quadratic decomposition of Appell sequences

Loureiro, Ana F.; Maroni, P.

doi:10.1016/j.exmath.2007.10.002

Cited by 17 publications

(18 citation statements)

References 6 publications

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“…Subsequently, regarding a more accurate information about the arisen G ε,µ -Appell sequences, in Section 3 we study these MPS through a functional point of view, where the range for the parameter µ was broadened to a subset of C (the set of complex numbers) satisfying (3.3). The unique D-Appell and F ε -Appell orthogonal sequences are, respectively, the Hermite (a result given by Angelescu [2] and later on by other authors [13,36] but further references may be found in [1]) and the Laguerre polynomial sequences of parameter ε/2, up to a linear change of variable (achieved in [28]). However, in Section 4 we conclude that a G ε,µ -Appell sequence cannot be orthogonal.…”

Section: 2)mentioning

confidence: 98%

“…Recently, in [28], the two authors have proceeded to the quadratic decomposition (hereafter QD) of an Appell polynomial sequence (that is, a MPS {B n } n⩾0 such that B n (·) = (n + 1) −1 B ′ n+1 (·), n ⩾ 0) [3]. The four associated sequences obtained by this approach are also Appell sequences but with respect to another differential operator:…”

Section: 2)mentioning

confidence: 98%

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Quadratic decomposition of Laguerre polynomials via lowering operators

Loureiro¹,

Maroni

2011

Journal of Approximation Theory

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A Laguerre polynomial sequence of parameter ε/2 was previously characterized in a recent work [Ana F. Loureiro and P. Maroni (2008) [28]] as an orthogonal F ε -Appell sequence, where F ε represents a lowering (or annihilating) operator depending on the complex parameter ε ̸ = −2n for any integer n ⩾ 0. Here, we proceed to the quadratic decomposition of an F ε -Appell sequence, and we conclude that the four sequences obtained by this approach are also Appell but with respect to another lowering operator consisting of a Fourth-order linear differential operator G ε,µ , where µ is either 1 or −1. Therefore, we introduce and develop the concept of the G ε,µ -Appell sequences and we prove that they cannot be orthogonal. Finally, the quadratic decomposition of the non-symmetric sequence of Laguerre polynomials (with parameter ε/2) is fully accomplished.

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Section: 2)mentioning

confidence: 98%

Section: 2)mentioning

confidence: 98%

Quadratic decomposition of Laguerre polynomials via lowering operators

Loureiro¹,

Maroni

2011

Journal of Approximation Theory

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“…Adopting a similar procedure as the one described in [10,Sect. 3], the dual sequence {u n } n≥0 associated to the M q;q −ε -Appell polynomial sequences {P n } n≥0 fulfills t M q;q −ε (u n ) = ρ n+1 u n+1 , n≥ 0.…”

Section: The Arisen L Q;εμ -Appell Polynomial Sequencementioning

confidence: 98%

Around q-Appell polynomial sequences

Loureiro

Maroni

2011

Ramanujan J

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First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator M q;q −ε , where ε represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of M √ q;q −ε/2 (the M √ q;q −ε/2 -Appell), only the Wall q-polynomials with parameter q ε/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.

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“…. (there is a vast literature on the quadratic decomposition of orthogonal polynomial sequences; see, for example, [22,26,27], where other references could be found, or [3, pp. 104ff.]). Their associated polynomials V (n) k are defined as above, and it holds that V (n)…”

Section: Adjacent Families and Quadratic Decompositionmentioning

confidence: 99%

Nonlinear functional equations satisfied by orthogonal polynomials

Brezinski

2010

Journal of Approximation Theory

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Let c be a linear functional defined by its moments c(x i ) = c i for i = 0, 1, . . .. We proved that the nonlinear functional equations P(t) = c(P(x)P(αx + t)) and P(t) = c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.

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Quadratic decomposition of Appell sequences

Cited by 17 publications

References 6 publications

Quadratic decomposition of Laguerre polynomials via lowering operators

Quadratic decomposition of Laguerre polynomials via lowering operators

Around q-Appell polynomial sequences

Nonlinear functional equations satisfied by orthogonal polynomials

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